A characterisation of Baer subplanes

S. G. Barwick; Wen-Ai Jackson

paper

A characterisation of Baer subplanes

arXiv:1906.04318

Abstract

Let $K$ be a set of $q^{2} + 2 q + 1$ points in $P G (4, q)$ . We show that if every 3-space meets $K$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $K$ is a ruled cubic surface. Moreover, $K$ corresponds via the Bruck-Bose representation to a tangent Baer subplane of $P G (2, q^{2})$ . We use this to prove a characterisation in $P G (2, q^{2})$ of a set of points $B$ as a tangent Baer subplane by looking at the intersections of $B$ with Baer-pencils.